Formulas for the conditional expectation that do not require computation of conditional distributions are desirable in practice. For example, the Tweedie formula for the conditional expectation \(\mathbb {E}[X\mid Y=y] = y + p_Y^{\prime }(y)/p_Y(y)\)  helped develop the empirical Bayes method [1]}. Similarly, the formula for the higher-order derivatives (REF ) might shed light on practical applications. For instance, one may obtain a uniform bound \(|(d^k/dy^k)\mathbb {E}[X\mid Y=y]|\le M^kk!\)  if, e.g., \(X\)  is bounded. This implies that the conditional expectation is real analytic. In particular, knowledge of the moments \(\mathbb {E}[X^\ell \mid Y=0]\)  (for \(\ell \in \mathbb {N}\) ) suffices to obtain \(\mathbb {E}[X \mid Y=y]\)  on the neighborhood \(y\in (-1/M,1/M)\)  via Taylor's expansion and the derivative expressions (REF ). Further, the value of the conditional expectation \(\mathbb {E}[X\mid Y=y]\)  over an interval \(y\in (\alpha ,\beta )\)  is retrievable by its evaluations at only \(\lceil M(\beta -\alpha )/2 \rceil +1\)  points.
